Prerequisites: Measurement and Integration, Functional Analysis.

Limited and unbounded linear operators. Integral operators, multiplication operators and differential operators. The extension theorem for bounded operators. The Fourier transform in L1 ( Rn ), S ( Rn ) and L2 ( Rn ). L. Schawartz distributions, tempered distributions and compact support distributions. Sobolev spaces Hs ( Rn ). Applications to linear and non-linear evolution equations. Closed, closable, symmetric and self-adjoint operators. Resolvent and spectrum. The Cayley transform. Differentiation of measures. Hahn’s decomposition theorem. The Radon-Nikodyn decomposition theorem. Riemann-Stieltjes and Lebesque-Stieltjes integrals. The spectral theorem for self-adjoint operators in the forms of spectral integrals, multiplication operators and functional calculus. Stone’s theorem.

References:
HILLE, E. – Methods in Classical and Functional Analysis. Reading, Mass., Addison-Wesley Pub. Co., 1972.
KOLMOGOROV, A. N., FOMIN, S. V. – Introductory Real Analysis, Dover Publ., Inc. Translated from the second Russian edition, 1970.
REED. M., BARRY, S. – Methods of Modern Mathematical Physics vols. I and II. New York : Academic Press, 1972-1978.
RIESZ, F., SZ. -NAGY, B. – Functional Analysis, Frederick Ungar Publ.Co. Translated from the second french edition, 1955.
RUDIN, W. – Real and Complex Analysis. New York, McGraw-Hill, 1966.
STONE, M. – Linear Transformations in Hilbert Space and their Applications to Analysis, Amer. Math. Soc. Colloq. Publ., vol. 15, 1932.
THAYER, J. – Self-Adjoint Operators and Partial Differential Equations. Rio de Janeiro, Projeto Euclides, IMPA, 1987.

* Basic syllabus. The teacher has the autonomy to make any changes.